A055786 -id:A055786 - cp's OEIS frontend

A162443 Numerators of the BG1[ -5,n] coefficients of the BG1 matrix.

5, 66, 680, 2576, 33408, 14080, 545792, 481280, 29523968, 73465856, 27525120, 856162304, 1153433600, 18798870528, 86603988992, 2080374784, 2385854332928, 3216930504704, 71829033058304, 7593502179328, 281749854617600

Offset: 1

Johannes W. Meijer, Jul 06 2009

The BG1 matrix coefficients are defined by BG1[2m-1,1] = 2*beta(2m) and the recurrence relation BG1[2m-1,n] = BG1[2m-1,n-1] - BG1[2m-3,n-1]/(2*n-3)^2 with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. . As usual beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). For the BG2 matrix, the even counterpart of the BG1 matrix, see A008956.
We discovered that the n-th term of the row coefficients can be generated with BG1[1-2*m,n] = RBS1(1-2*m,n)* 4^(n-1)*((n-1)!)^2/ (2*n-2)! for m >= 1. For the BS1(1-2*m,n) polynomials see A160485.
The coefficients in the columns of the BG1 matrix, for m >= 1 and n >= 2, can be generated with GFB(z;n) = ((-1)^(n+1)*CFN2(z;n)*GFB(z;n=1) + BETA(z;n))/((2*n-3)!!)^2 for n >= 2. For the CFN2(z;n) and the Beta polynomials see A160480.
The BG1[ -5,n] sequence can be generated with the first Maple program and the BG1[2*m-1,n] matrix coefficients can be generated with the second Maple program.
The BG1 matrix is related to the BS1 matrix, see A160480 and the formulas below.

			The first few formulas for the BG1[1-2*m,n] matrix coefficients are:
BG1[ -1,n] = (1)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -3,n] = (1-2*n)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!
The first few generating functions GFB(z;n) are:
GFB(z;2) = ((-1)*(z^2-1)*GFB(z;1) + (-1))/1
GFB(z;3) = ((+1)*(z^4-10*z^2+9)*GFB(z;1) + (-11 + z^2))/9
GFB(z;4) = ((-1)*( z^6- 35*z^4+259*z^2-225)*GFB(z;1) + (-299 + 36*z^2 - z^4))/225

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.
J. M. Amigo, Relations among Sums of Reciprocal Powers Part II, International Journal of Mathematics and Mathematical Sciences , Volume 2008 (2008), pp. 1-20.

A162444 are the denominators of the BG1[ -5, n] matrix coefficients.
The BG1[ -3, n] equal (-1)*A002595(n-1)/A055786(n-1) for n >= 1.
The BG1[ -1, n] equal A046161(n-1)/A001790(n-1) for n >= 1.
The cs(n) equal A046161(n-2)/A001803(n-2) for n >= 2.
The BETA(z, n) polynomials and the BS1 matrix lead to the Beta triangle A160480.
The CFN2(z, n), the t2(n, m) and the BG2 matrix lead to A008956.
Cf. A000364, A001818 and A160485.
Cf. A162443 (BG1 matrix), A162446 (ZG1 matrix) and A162448 (LG1 matrix).

a := proc(n): numer((1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!) end proc: seq(a(n), n=1..21);
# End program 1
nmax1 := 5; coln := 3; Digits := 20: mmax1 := nmax1: for n from 0 to nmax1 do t2(n, 0) := 1 od: for n from 0 to nmax1 do t2(n, n) := doublefactorial(2*n-1)^2 od: for n from 1 to nmax1 do for m from 1 to n-1 do t2(n, m) := (2*n-1)^2* t2(n-1, m-1) + t2(n-1, m) od: od: for m from 1 to mmax1 do BG1[1-2*m, 1] := euler(2*m-2) od: for m from 1 to mmax1 do BG1[2*m-1, 1] := Re(evalf(2*sum((-1)^k1/(1+2*k1)^(2*m), k1=0..infinity))) od: for m from -mmax1 +coln to mmax1 do BG1[2*m-1, coln] := (-1)^(coln+1)*sum((-1)^k1*t2(coln-1, k1)*BG1[2*m-(2*coln-1)+2*k1, 1], k1=0..coln-1)/doublefactorial(2*coln-3)^2 od;
# End program 2
# Maple programs edited by Johannes W. Meijer, Sep 25 2012

a(n) = numer(BG1[ -5,n]) and A162444(n) = denom(BG1[ -5,n]) with BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!.
The generating functions GFB(z;n) of the coefficients in the matrix columns are defined by
GFB(z;n) = sum(BG1[2*m-1,n]*z^(2*m-2), m=1..infinity).
GFB(z;n) = (1-z^2/(2*n-3)^2)*GFB(n-1) - 4^(n-2)*(n-2)!^2/((2*n-4)!*(2*n-3)^2) for n => 2 with GFB(z;n=1) = 1/(z*cos(Pi*z/2))*int(sin(z*t)/sin(t),t=0..Pi/2).
The column sums cs(n) = sum(BG1[2*m-1,n]*z^(2*m-2), m=1..infinity) = 4^(n-1)/((2*n-2)*binomial(2*n-2,n-1)) for n >= 2.
BG1[2*m-1,n] = (n-1)!^2*4^(n-1)*BS1[2*m-1,n]/(2*n-2)!

A002595 Denominators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).

Original entry on oeis.org

1, 6, 40, 112, 1152, 2816, 13312, 10240, 557056, 1245184, 5505024, 12058624, 104857600, 226492416, 973078528, 2080374784, 23622320128, 30064771072, 635655159808, 446676598784, 11269994184704, 23639499997184, 6597069766656

Offset: 0

N. J. A. Sloane

arcsin(x) is usually written as x + x^3/(2*3) + 1*3*x^5/(2*4*5) + 1*3*5*x^7/(2*4*6*7) + ..., = x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ... when reduced to lowest terms.
arccos(x) = Pi/2 - (x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ...).
arccsc(x) = 1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+...
arcsec(x) = Pi/2 -(1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+...)
arcsinh(x) = x-1/6*x^3+3/40*x^5-5/112*x^7+35/1152*x^9-63/2816*x^11+...
arccsc(x) = arcsin(1/x) and arcsec(x) = arccos(1/x): 1 < |x|
arccsch(x) = arcsinh(1/x) for 1 < |x|
Also denominator of (2n-1)!! / ((2n+1)*(2n)!!) (n=>0).

W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, Chap. 3.
Focus, vol. 16, no. 5, page 32, Oct 1996.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 31, equation 31:6:1 at page 290.

T. D. Noe, Table of n, a(n) for n=0..200
H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.
Eric Weisstein's World of Mathematics, Inverse Cosecant.
Eric Weisstein's World of Mathematics, Inverse Cosine.
Eric Weisstein's World of Mathematics, Inverse Secant.
Eric Weisstein's World of Mathematics, Inverse Sine.
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosecant.
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Sine.
Eric Weisstein's World of Mathematics, Archimedes' Spiral.
Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.

A055786(n) / a(n) = A001147(n) / ( A000165(n) * (2*n+1))
Cf. A162443 where BG1[ -3,n] = (-1)*A002595(n-1)/A055786(n-1) for n =>1. - Johannes W. Meijer, Jul 06 2009
a(n) = 2*A143582(n+1) for n>=1. - Filip Zaludek, Oct 25 2016

Denominator[Take[CoefficientList[Series[ArcSin[x],{x,0,50}],x],{2,-1,2}]] (* Harvey P. Dale, Aug 06 2012 *)

a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))); \\ Stefano Spezia, Dec 31 2024

a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))). - Johannes W. Meijer, Jul 06 2009

A052468 Numerators in the Taylor series for arccosh(x) - log(2*x).

Original entry on oeis.org

1, 3, 5, 35, 63, 77, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 646323, 300540195, 583401555, 756261275, 4418157975, 6892326441, 22427411435, 263012370465, 514589420475, 2687300306925, 15801325804719, 61989816618513, 121683714103007

Offset: 1

Eric W. Weisstein

A055786 is the preferred version of this sequence.

			i*Pi/2 - arccosh(x) = i*x + (1/6)*i*x^3 + (3/40)*i*x^5 + (5/112)*i*x^7 + (35/1152)*i*x^9 + (63/2816)*i*x^11 + (231/13312)*i*x^13 + (143/10240)*i*x^15 + (6435/557056)*i*x^17 + ...
0, 1, 0, 1/6, 0, 3/40, 0, 5/112, 0, 35/1152, 0, 63/2816, 0, 231/13312, 0, 143/10240, 0, 6435/557056, 0, 12155/1245184, 0, 46189/5505024, 0, ... = A052468/A052469

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Secant
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosecant
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosine
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Sine

See A055786 for further information.
a(n)/A052469(n) = (1/(2*n))*A001790(n)/A046161(n) for n=>1.
Equals A162441(n+1)/(2n+1) for n=>1. - Johannes W. Meijer, Jul 06 2009

List([1..30], n-> NumeratorRat( Factorial(2*n-1)/(4^n*(Factorial(n))^2) )) # G. C. Greubel, May 18 2019

[Numerator(Factorial(2*n-1)/( 2^(2*n)* Factorial(n)^2)): n in [1..30]]; // Vincenzo Librandi, Jul 10 2017

a [n_]:=Numerator[(2 n - 1)! / (2^(2 n) n!^2)]; Array[a, 40] (* Vincenzo Librandi, Jul 10 2017 *)

{a(n) = numerator((2*n-1)!/(4^n*(n!)^2))}; \\ G. C. Greubel, May 18 2019

[numerator(factorial(2*n-1)/(4^n*(factorial(n))^2)) for n in (1..30)] # G. C. Greubel, May 18 2019

a(n)/A052469(n) = A001147(n)/(A000165(n)*2*n). E.g., a(6) = 77 = 1*3*5*7*9*11 / gcd( 1*3*5*7*9*11, 2*4*6*8*10*12*12 ).
a(n) = numerator((2*n-1)!/(4^n * (n!)^2)). - Johannes W. Meijer, Jul 06 2009
Let z(n) = 2*(2*n+1)!*4^(-n-1)/((n+1)!)^2, then a(n) = numerator(z(n)), A162442(n) = denominator(z(n)), and z(n) = 1/(n+1) - Sum_{k=0..n}(-1)^k*binomial(n,k)*z(k). - Groux Roland, Jan 04 2011
a(n) = numerator(binomial(2n,n)/(n*2^(2n-1))). - Daniel Suteu, Oct 30 2017

Updated by Frank Ellermann, May 22 2011

Cross-references edited by Johannes W. Meijer, Jul 05 2009

A101681 Numbers k such that gcd(C(2k,k), 2k+1) > 1.

Original entry on oeis.org

7, 16, 17, 19, 22, 25, 31, 34, 38, 42, 43, 45, 46, 47, 49, 52, 55, 58, 61, 64, 67, 70, 71, 72, 73, 76, 77, 79, 80, 82, 87, 88, 92, 93, 94, 97, 100, 102, 103, 104, 106, 107, 110, 112, 115, 117, 122, 123, 124, 127, 129, 130, 133, 136, 139, 142, 143, 145, 147, 148

Offset: 1

Ralf Stephan, Dec 11 2004

nonn

Positions where A056616 differs from A000984.
The set seems to have greater cardinality than its complement.
Positions where A055786 differs from A001790. - Mohammed Yaseen, Aug 03 2024

			7 is in the sequence as gcd(binomial(2*7, 7), 2*7 + 1) = gcd(3432, 15) = gcd(3*1144, 3*5) > 1. - _David A. Corneth_, Apr 03 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000984, A056616, A126786.

Select[Range[200],GCD[Binomial[2 #,#],2 #+1]>1&] (* Harvey P. Dale, May 11 2019 *)

is(n) = { my(f = factor(2*n+1)); for(i = 1, #f~, if(val(2*n, f[i, 1])-2*val(n, f[i, 1]) > 0, return(1))); 0 }
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Apr 03 2021

A143581 Numerators of coefficient of x^(n+1/2) in the series expansion of the haversine.

Original entry on oeis.org

2, 1, 3, 5, 35, 63, 231, 143, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 9694845, 100180065, 116680311, 2268783825, 1472719325, 34461632205, 67282234305, 17534158031, 514589420475, 8061900920775, 5267108601573

Offset: 1

Eric W. Weisstein, Aug 24 2008

			2*sqrt(z) + z^(3/2)/3 + (3*z^(5/2))/20 + (5*z^(7/2))/56 + (35*z^(9/2))/576 + ...

Eric Weisstein's World of Mathematics, Inverse Haversine

Cf. A143582.

Cf. A055786. - R. J. Mathar, Sep 05 2008

A162444 Denominators of the BG1[ -5,n] coefficients of the BG1 matrix.

Original entry on oeis.org

1, 1, 3, 5, 35, 9, 231, 143, 6435, 12155, 3553, 88179, 96577, 1300075, 5014575, 102051, 100180065, 116680311, 2268783825, 210388475, 6892326441, 67282234305, 17534158031, 39583801575, 8061900920775, 169906729083

Offset: 1

Johannes W. Meijer, Jul 06 2009

For the numerators of the BG1[ -5,n] coefficients see A162443.

We observe that BG1[ -3,n] = (-1)*A002595(n-1)/A055786(n-1), i.e. they equal the inverted coefficients of the series expansion of arcsin(x), and that BG1[ -1,n] = A046161(n-1)/A001790(n-1), i.e. they equal the inverted coefficients of the series expansion of 1/sqrt(1-x).

			The first few formulas for the BG1[1-2*m,n] matrix coefficients are:
BG1[ -1,n] = (1)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -3,n] = (1-2*n)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -7,n] = (1-2*n+60*n^2-120*n^3)*4^(n-1)*(n-1)!^2/(2*n-2)!

A162443 are the numerators of the BG1[ -5, n] matrix coefficients.
The BG1[ -3, n] equal A002595(n-1)/A055786(n-1) for n =>1.
The BG1[ -1, n] equal A046161(n-1)/A001790(n-1) for n =>1.

a(n) = denom(BG1[ -5,n]) and A162443(n) = numer(BG1[ -5,n]) with BG1[ -5,n] = 4^(n-1)*(1-8*n+12*n^2)*(n-1)!^2/ (2*n-2)!.

A091019 Denominators of the Taylor series of arccosh(z)/sqrt(2(x-1)) about 1.

Original entry on oeis.org

1, 12, 160, 896, 18432, 90112, 851968, 1310720, 142606336, 637534208, 5637144576, 24696061952, 429496729600, 1855425871872, 15942918602752, 68169720922112, 1548112371908608, 3940649673949184, 166633186212708352

Offset: 0

Eric W. Weisstein, Dec 13 2003

nonn

Numerators are A055786.

			Arccosh(z) = 1 + (1 - z)/12 + (3*(-1 + z)^2)/160 - (5*(-1 + z)^3)/896 + (35*(-1 + z)^4)/18432 - (63*(-1 + z)^5)/90112 + ...

Q. Wei, R. Onofrio, Edge effects in electrostatic calibrations for the measurement of the Casimir force, Phys. Lett. A 374 (2010) 2230-2234.
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosecant

Cf. A055786.

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A162443 Numerators of the BG1[ -5,n] coefficients of the BG1 matrix.

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A002595 Denominators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).

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A052468 Numerators in the Taylor series for arccosh(x) - log(2*x).

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A101681 Numbers k such that gcd(C(2k,k), 2k+1) > 1.

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A143581 Numerators of coefficient of x^(n+1/2) in the series expansion of the haversine.

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A162444 Denominators of the BG1[ -5,n] coefficients of the BG1 matrix.

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A091019 Denominators of the Taylor series of arccosh(z)/sqrt(2(x-1)) about 1.

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